The Boussinesq problem in dipolar gradient elasticity

Abstract

The three-dimensional axisymmetric Boussinesq problem of an isotropic half-space subjected to a concentrated normal quasi-static load is studied within the framework of linear dipolar gradient elasticity. Our main concern is to determine possible deviations from the predictions of classical linear elastostatics when a more refined theory is employed to attack the problem. Of special importance is the behavior of the new solution near to the point of application of the load where pathological singularities exist in the classical solution. The use of the theory of gradient elasticity is intended here to model the response of materials with microstructure in a manner that the classical theory cannot afford. A linear version of this theory results by considering a linear isotropic expression for the strain-energy density that depends on strain-gradient terms, in addition to the standard strain terms appearing in classical elasticity. Through this formulation, a microstructural material constant is introduced, in addition to the standard Lamé constants. The solution method is based on integral transforms and is exact. The present results show significant departure from the predictions of classical elasticity. Indeed, continuous and bounded displacements are predicted at the points of application of the concentrated load. Such a behavior of the displacement field is, of course, more natural than the singular behavior exhibited in the classical solution.---

The Hertz contact problem in couple-stress elasticity

“…In fact, the couple-stress elasticity theory has been already employed successfully to model size effects in fracture (Gourgiotis and Georgiadis, 2007;Radi, 2008;Mishuris et al, 2012;Morini et al, 2013) and contact problems (Zisis et al, 2014;Zisis, 2017;Karuriya and Bhandakkar, 2017;Song et al, 2017). It should be further noted that only few three dimensional solutions exist within the context of gradient theories with the majority of them concerning traction boundary value problems for concentrated or distributed loads (see for example Ejike, 1970;Anagnostou et al, 2013;Gao and Zhou, 2013;Georgiadis et al, 2014);…”

The Cerruti problem in dipolar gradient elasticity

“…It was found that the (exact) gradient solution predicts bounded and continuous displacements at the point of application of the load and, therefore, ‘corrects’ (in a boundary-layer sense) the classical Flamant–Boussinesq solution (the latter one predicts a logarithmically singular normal displacement and a discontinuous tangential displacement in the vicinity of the point of application of the concentrated line load). In addition, an exact solution for the 3D axisymmetric Boussinesq problem was given recently by the present authors (Georgiadis et al [14]), showing that the Cauchy-type singularities in the displacement components are eliminated in the context of the dipolar gradient elasticity. Such a behaviour is, of course, more natural than the singular behaviour present in the classical solution.…”

“…Notice that use of the Hankel transform is not appropriate, since the problem is non-axisymmetric . It is also remarked that the rest of this section essentially follows the general analysis from our recent related work regarding the 3D Boussinesq problem in dipolar gradient elasticity (Georgiadis et al [14]). The analysis is briefly presented here for the sake of completeness and because of the need to introduce certain definitions.…”

“…Moreover, the surface displacement normal to the direction of the concentrated load and the surface vertical displacement both vanish at the point (origin of the coordinate system) where the load is applied. As was pointed out by Georgiadis et al [14], the occurrence of bounded displacements at the point of application of the concentrated load implies that the total strain energy in a small region surrounding the singular point vanishes, as the size of the region tends to zero. Consequently, an analogue of Kirchhoff’s theorem guarantees uniqueness of solution for the concentrated load problem in gradient elasticity (Grentzelou and Georgiadis [33]).…”

Gradient-enriched finite element methodology for axisymmetric problems

“…In particular, strain-gradient theories such as the Mindlin theory have been successfully applied to study different axisymmetric problems such as stress concentrations at spherical inclusions and cavities [21,25], circular holes [32,33], Boussinesq problem [35,36]. Other researchers, instead, provided analytical solutions to axisymmetric problems such as thick-walled cylinder [34], annulus under internal and external pressure, and circular hole in infinite body [12] starting from the gradient elastic formulation proposed by Aifantis et al [1,10,11].…”